A067912
Engel expansion of zeta(4) = Pi^4/90 = Sum_{i>0} 1/i^4.
Original entry on oeis.org
1, 13, 15, 19, 132, 1474, 1977, 10392, 12992, 44777, 59412, 170685, 217607, 704791, 818133, 1387423, 2208674, 3206215, 12732462, 13962681, 24593168, 39744274, 55804517, 130269696, 426536424, 546807194, 1030799587, 1139987135
Offset: 1
See
A006784 for explanation of Engel expansions.
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EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]]; EngelExp[N[Pi^4/90, 7!], 20] (* G. C. Greubel, Dec 26 2016 *)
A091833
Pierce expansion of 1/zeta(2).
Original entry on oeis.org
1, 2, 4, 7, 22, 29, 51, 173, 210, 262, 417, 746, 12341, 207220, 498538, 1286415, 2351289, 3702952, 7664494, 54693034, 75971438, 269954954, 6674693008, 13449203581, 59799655308, 98912303039, 948887634688, 3557757020909, 5898230078743
Offset: 1
- G. C. Greubel, Table of n, a(n) for n = 1..1001 [a(1)=1 inserted by Georg Fischer, Nov 20 2020]
- P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Théor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
- Vlado Keselj, Length of Finite Pierce Series: Theoretical Analysis and Numerical Computations .
- Jeffrey Shallit, Some predictable Pierce expansions, Fib. Quart., 22 (1984), 332-335.
- Pelegrí Viader, Lluís Bibiloni, Jaume Paradís, On a problem of Alfred Renyi, Economics Working Paper No. 340.
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PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[1/Zeta[2], 7!], 25] (* G. C. Greubel, Nov 14 2016 *)
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default(realprecision, 100000); r=zeta(2); for(n=1, 100, s=(r/(r-floor(r))); print1(floor(r), ", "); r=s) \\ Benoit Cloitre [amended by Georg Fischer, Nov 20 2020]
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